Introduction

The availability of mineral resources and the technological capacity to obtain and use them are two primary determinants of a nation’s prosperity and are essential for our society’s development (Bascompta et al. 2022). However, the improper exploitation of raw minerals can cause significant environmental issues such as water pollution (El Khalil et al. 2008), soil contamination (Montalván-Olivares et al. 2021), loss of wild biodiversity (Murguía et al. 2016) or subsidence (Hamdi et al. 2018; Mason et al. 2021). Consequently, there is a need for new methodologies to improve mining practices and reduce environmental impact, as seen in initiatives like the Responsible Research Innovation (RRI) from Horizon 2020, which aims to make industries more inclusive and sustainable (Iatridis and Schroeder 2016). In addition, responsible mining has a positive impact on the economic growth of the mining industry (Yousefian et al. 2024).

Underground mining is particularly associated with subsidence, affecting both mining infrastructure and inhabited areas (Pipia et al. 2007; Perissin and Wang 2011; Solarski et al. 2022). Subsidence can be classified as a direct or indirect process; on the one hand, direct processes such as post-mining voids to the surface; on the other hand, mining-induced dewatering, which in some mining operations is a key aspect and has typically been understudied (Guzy and Witkowski 2021). Moreover, other indirect processes can be found in subsidence due to mining-induced earthquakes (Malinowska et al. 2018; Witkowski et al. 2024). Both types of subsidence can be related to the extraction of essential mineral resources, such as groundwaters or potash (Sanmiquel et al. 2018; Figueroa-Miranda et al. 2018).

The main use of potash ore is as a fertiliser, and nowadays, its use is essential for the continued production of crops (Zörb et al. 2014). Therefore, it can be considered a vital ore in reducing world hunger. For example, more than 828 million people worldwide suffered from hunger in 2021 (FAO et al. 2022). Moreover, the United Nations stated in 2022 that 17 Sustainable Development Goals (SDG), especially the second SDG, “Zero Hunger”, and the fifteenth “Life on Land”, are completely linked to potash extraction. Potash extraction, often through deep underground mining, poses significant environmental challenges, especially related to subsidence (Broughton 2019; Ushakova et al. 2023). Several environmental impacts on the surface of the mine can be developed due to the mining works (Warren 2016; Ushakova et al. 2023). It is well-known that the main impact of the construction of underground potash ore deposits involves geotechnical challenges, mainly related to time-dependent behaviour (Campos de Orellana 1996; Corthésy et al. 2003; Marketos et al. 2015; Minkley et al. 2016; Yubero et al. 2021) and subsidence processes (Yerro et al. 2014; Modeste et al. 2021). The effects of subsidence processes appear mostly on the surface, such as cracks, changes in the topography of the ground or even collapses (Baryakh et al. 2016; Baryakh and Samodelkina 2018). Although deposits deeper than 1,000 m are not common, there are some examples, such as the Saskatchewan ore deposit (Canada), which is one of the unique examples of long-term potash mining at such depths (Van Sambeek 1997; Ong et al. 2007; Samsonov et al. 2014; Baryakh et al. 2021). Nowadays, several deep potash mines are under construction or have already been built in countries like Russia or Spain (Baryakh et al. 2015, 2021; Sanmiquel et al. 2018; Sidki-Rius et al. 2022), representing a challenge to reconcile their exploitation with the needs of the surrounding ecosystem. Numerous studies have investigated subsidence phenomena and their environmental implications, primarily focusing on subsidence basins due to soft ground tunnelling or coal mining. One of the earliest geotechnical studies on ground subsidence in clay soils was conducted by Terzaghi and Peck (1948). In subsequent years, Schmidt (1969) examined theories and methods to predict ground movement from soft ground tunnelling. Clough and Schmidt (1981) discussed geomechanical behaviour in soft clay excavations and tunnels. O’Reilly and New (1982) reviewed settlement and ground movement measurements in UK tunnelling projects covering various soil types. Rankin (1988) guided the estimation of the effects of tunnel construction in urban areas with soft soils, including empirical approaches for defining subsidence zones, assessing surface movement, and proposing risk classifications.

Coal mining has historically caused subsidence, as has the extraction of metalliferous ores, critical raw materials, and overexploitation of groundwaters (Behera and Rawat 2023). Key works are reviewed to understand subsidence management better. In 1975, the National Coal Board developed a method for characterising subsidence basin parameters after analysing several coal ore deposits in the United Kingdom. Kratzsch (1983) presented the effects of surface ground and shaft damage due to mining, outlining the basis of knowledge of ground movement at that time. Using an integrated approach, Peng (1992) defined and determined subsidence basin parameters by analysing 110 cases from major US coal deposits. Garrett (1996) highlighted the common techniques and risks in potash mining, emphasising the need for case-specific analysis. Sheorey et al. (2000) analysed discontinuous subsidence processes in Indian coalfields using the influence function methodology. Toraño et al. (2000) used the profile function methodology to predict subsidence from steep coal seam exploitation. Yan et al. (2021a) applied theoretical analysis to study surface subsidence boundaries due to horizontal coal seam mining. As previously stated, other examples of subsidence caused by the extraction of economically valuable ores, such as metalliferous or critical raw materials, can be found in works like Contrucci et al. (2019), where the post-mining ground risk was assessed in an iron ore deposit located in France. The overburden of the area was found to be completely faulted, which makes the monitoring inefficient. Finally, GNSS technology was identified as the best to monitor the area. Furthermore, in the study of Murguía and Bringezu (2016), a novel methodology was presented to measure the cumulative area disturbed based on analysis of satellite images. The authors analyzed several ore deposits, including Critical Raw Materials (CRM), such as gold, silver, copper bauxite and iron. Finally, focusing on subsidence caused by overexploitation of groundwater, Abidin et al. (2008) developed a study about the characteristics of land subsidence caused by overexploitation of groundwater resources InSAR and GPS technology. Their research suggests a framework for sustainable subsidence monitoring that shares the same satellite methodology as the current study. Other examples can be found in the studies done in the city of Calcutta, where the overexploitation of groundwaters produced subsidence of 11 mm/year in the south part of the city, and the vicinities of the metropolitan area subsidence rate reached values of 5 to 6 mm/year (Behera and Rawat 2023; Chatterjee et al. 2006).

In recent years, research on infrastructure-related subsidence processes, such as tunnels and roads, has increasingly utilised technologies like Interferometric Synthetic Aperture Radar (InSAR) and the Global Navigation Satellite System (GNSS). Yan et al. (2021b) examined subsidence impacts from tunnelling at the Beijing-Zhangzhou railway, highlighting InSAR’s role in analysing subsidence in soft clay. Bonì et al. (2015) monitored a severe subsidence process over 20 years in the Alto Guadalentín area, Spain, using DInSAR techniques due to aquifer overexploitation. Bitelli et al. (2000) proposed a levelling network linked to a GPS network to monitor subsidence in the southern Po Valley, where anthropogenic activities increased subsidence rates. Mancini et al. (2009) assessed subsidence in Tuzla, Bosnia Herzegovina, due to solution mining of a salt deposit, finding a correlation between subsidence rate and salt mining. Buzzanga et al. (2020) analysed subsidence in Hampton Roads, Virginia, using a combination of InSAR and GPS.

Despite several contributions to the field of land subsidence, there is a lack of understanding regarding subsidence processes in geological environments like potash ore deposits. For example, Rucker et al. (2013) identified three subsidence zones using InSAR following a brine well collapse in New Mexico but did not quantify subsidence basin parameters. Baryakh et al. (2021) studied subsidence from deep potash mining in Russia using the Finite Element Method (FEM) to improve boundary angle values, though combining FEM with InSAR and GPS data could enhance accuracy (Sidki-Rius et al. 2022). This highlights the importance of researching surface subsidence in potash mining and underscores the need for accurate subsidence profiles and parameters for effective land management.

This study characterises the parameters governing subsidence in the Catalan Potash Basin (CPB) from 1995 to 2021, aiming to improve land subsidence management. The methodology employs advanced remote sensing techniques, specifically InSAR and GPS, to provide a detailed analysis and understanding of the subsidence processes over time. Furthermore, this research calculates and defines subsidence profiles in the study area, which leads to identifying and quantifying subsidence parameters, including boundary angle and the distance of influence (Knothe 1957; National Coal Board 1975). In addition, the characteristic subsidence function has been approximated to the Gaussian function using the least squares methodology.

Case study

The Catalan Potassium Basin (CPB) is located in the Ebro Basin, Spain (Cendón et al. 2003). The CPB is 150 km2, and it can be subdivided into east and west sides regarding the main ore deposits, namely W-CPB and E-CPB, respectively. The case study was carried out based on data from the east side. Figure 1 consists of two complementary sections, a satellite image showing the Ebro Basin and the Catalan Potash Basin (CPB), which is further divided into eastern and western segments as previously noted. Additionally, a geological profile (A-A’) is provided to enhance the map’s representation, illustrating the characteristic folded structures of the region.

All the layers of the deposit are stratified with the presence of clay minerals. The layers forming the deposit are bent due to the tectonic forces experienced during the Alpine orogeny and the well-known ductility of the salt materials (Campos de Orellana 1996). Figure 2 shows the stratigraphic column of the main ore deposit of E-CPB. The mining zone has two different mineable layers (layers A and B), composed of sylvite and rock salt in between with an intermediate layer consisting of salt (see Figs. 2 and 3). The deposit is mined using the room and pillar method, with extraction depths varying from 500 to 700 m. The average profile size of the mining drifts is from 6 m to 15 m in height and a width size ranging from 8 m to 10 m, with an average horizontal distance between tunnels of 9 m, in case of a salt layer between A and B being thicker than 5 m, two drifts are excavated to extract the potash ore, this type of mining design can be located in the northern part of the ore deposit (see Fig. 3). The extraction rate, depending on the arrangement of the layers, ranges between 60% and 70%. The average mineral excavation rate from 2015 to 2020 was 3 million tons.

Fig. 1
figure 1

The satellite image highlights the Ebro Basin in yellow and the Catalan Potash Basin in red. The case study is located on the eastern side of CPB (E-side). The white line indicates the direction of the geological profile (A-A’) depicted below the satellite image (according to Vergés 1999)

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Fig. 2
figure 2

Stratigraphic column of the study area (based on Campos de Orellana 1996)

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Fig. 3
figure 3

Mine design options depend on the arrangement of the ore layers (after Sanmiquel et al. 2018)

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Methods and materials

The following three sections will describe the methodology used. Firstly, the database was created based on GNSS and InSAR techniques. Following this, a specific methodology was designed based on CAD software coupled with the abovementioned methods. The definition of nine sections allowed the study of 74 subsidence profiles, which provided an accurate analysis of the selected area. Finally, a hybrid methodology combining the methods proposed by the National Coal Board (1975) and the approximation of the subsidence curve by a Gaussian distribution using the least squares method has been successfully applied.

Database creation

An analysis was carried out to identify the typical values defining the surface of a subsidence basin, as well as the angles and the governing function. An analysis was carried out to identify the typical values defining the surface of a subsidence basin, as well as the angles and the governing function. The total value of surface displacements is believed to happen due to the propagation of post-mining void; in the study area, there is no evidence that the subsidence process is linked to an indirect impact, such as rock mass drainage. The model was developed using classical equations defining surface subsidence parameters (National Coal Board 1975). For this purpose, the topographic characteristics of the terrain were considered using measurements carried out over non-consecutive twelve years from 1995 to 2021. In Table 1, a correlation between periods and techniques employed is shown. The case study has been monitored over an area of 46 km2 using GNSS and InSAR technology. The set of GNSS control points has been used to cover the whole area affected by the subsidence process. Additionally, InSAR imagery has been used to complement the monitoring system. Although some urban regions exist, the target area is dominated by forest and agricultural land. GNSS and InSAR methods were employed due to their reliability and accuracy in mining subsidence research (Amelung et al. 1999; Rucker et al. 2013; Diao et al. 2019; Modeste et al. 2021; Babayants et al. 2023). Only in the first periods, 1995 to 2003 and 2003 to 2008, were the measurement campaigns developed by the mining company, and the methodology used was classic total station topography. In 2008, the data was transferred to the research group, and a new methodology was established using GNSS; in 2016, in collaboration with an external company specialised in Interferometric technology, InSAR was coupled to GNSS as a new subsidence monitoring method.

The GNSS method used is based on static differential GPS with dual frequency receivers, using four devices. This way, two are considered bases, placing them in two well-identified coordinate points. The other two devices are used to measure the control points. The minimum measurement time for each control point is 12 min. Thus, for each point, it was possible to obtain the coordinates in the three axes, X, Y, and Z, with a Root Mean Square Error (RMSE) of two centimetres, taking into account that the methodology used has an accuracy of one centimetre in planimetric and altimetric coordinates. In addition, a double quality control has been performed with the following steps: Firstly, during the GPS post-process, using the Magnet Tools software (version 6.1.0.), a warning is set off when an error higher than two cm is detected when the error is detected, the point is remeasured twice within a one-week gap. Secondly, some points are randomly selected to be remeasured periodically to control if there is any problem with the measurement.

InSAR technology combined with GNSS points has an associated RMSE of two centimetres, making it a well-established and reliable method (Sanmiquel et al. 2018). The given approach used an average of 176 InSAR images from the SENTINEL-1 satellite coupled with an average of 241 GNSS points; details are shown in Table 1.

The deployment of InSAR with GNSS data in a Geographic information system (GIS) software provides a digital vertical displacement model for the whole period. Consequently, utilizing the GIS software’s curvature analysis functionality, a detailed subsidence surface for each period is feasible (Fig. 4).

Fig. 4
figure 4

Methodology proposed to calculate the subsidence surface

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Table 1 Correlation between periods and methodologies used
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Database management

AutoCAD and TCP-MDT software were used to analyse the subsidence surfaces. Nine sections were selected in the area affected by subsidence (Fig. 5); four of them cross the target area from East to South West, identified with numbers (1 to 4), while five vertical sections from North to South (A to E). A Metric Point (MP) is considered every 10 m in all sections of the twelve periods, showing the displacement in the Z-coordinate. The calculation periods used to determine each subsidence base are accumulative, using more than five years, which corresponds to 90% of the subsidence that can be formed in the case study (Sanmiquel et al. 2018) and, therefore, it may be possible to detect a well-formed subsidence base in all of them and see how progresses. According to nine sections and the displacement surface for each indicated period, 74 subsidence basin profiles were analysed.

Fig. 5
figure 5

The mining infrastructure built from 2008 to 2020 is shown in red, and the nine sections are pink

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The proposed methodology allows the calculation of the distance of influence by establishing the start and end parts of each subsidence profile and the total depth for each subsidence profile. In Fig. 6, accumulative subsidence basin profiles of Sect. 01 are shown; each colour indicates a different period to observe its evolution over time; it can be noticed that the subsidence profile becomes better defined as the period increases, since with time, mining infrastructure increases. The maximum subsidence recorded is 325 cm with an inflection point of 150 cm. Although the maximum subsidence depth ranges between 100 cm and 325 cm for the 74 profiles mentioned, they all follow a similar proportion, with their inflexion point around 40–50% of their maximum subsidence. Considering the RMSE for both techniques used, the accuracy of the subsidence basin profiles can be confirmed. Mining subsidence is a phenomenon that is closely related to the mining excavation ratio, among other geological and mining parameters (Hunt 1980; Salmi et al. 2017; Sasaoka et al. 2015; Diao et al. 2019). Points in red indicate zero subsidence (start and end point of subsidence basin). Taking these points and the mining map into account from 1995 to 2021, it was possible to determine the distance of influence, which is the shortest distance between the point of zero subsidence and the nearest mining drift, allowing the determination of the distance of influence for all subsidence profiles. However, to calculate the total depth from the surface to the drift, it was necessary to check the cartographic maps available from the Cartographic and Geological Institute of Catalonia (ICGC) since the depths indicated on the mining map are referenced to sea level.

The boundary angle is defined by the zero-subsidence point and the total depth of the mining drifts. Considering this definition, it can be calculated following the mathematical relationship stated by the National Coal Board (1975), as shown in Fig. 7. Determining the characteristic boundary angle for the area of interest was done through statistical calculation. Finally, Fig. 8 shows the methodology used up to the reaching point of the characteristic boundary angle and distance of influence.

Fig. 6
figure 6

Example of 11 subsidence basins from 2003 to 2021

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Fig. 7
figure 7

Mathematic relationship scheme

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Fig. 8
figure 8

Methodology proposed to calculate the boundary angle and the distance of influence

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Characteristic function of the subsidence basin

The effect of underground mining drifts on the surface topography is inevitably associated with the resulting ground movements in a subsidence basin. A considerable amount of data is available from field measurements of surface settlement profiles on tunnels in clays. Figure 9 has been used to summarise the settlement trough adopted from several research, where the surface vertical settlements and horizontal stress and displacement are shown (Schmidt 1969; Peck 1969; National Coal Board 1975; Clough and Schmidt 1981; O´Reilly and New 1982; Rankin 1988; Peng 1992). The green field settlement profile, which can represent the profile of a subsidence basin over a single tunnel, can generally be approximated by the error function or normal probability curve (also known as the Gaussian curve) as follows:

$$\:{S}_{v}\left(x\right)={S}_{vmax}\;exp\;exp\:\left(\frac{-{x}^{2}}{2\:{i}_{x}^{2}}\right)\:$$
(1)

where Svmax is the maximum surface subsidence at the centre line of the tunnel or drift, Sv is the surface subsidence at displacement distance x from the tunnel centre line, x is the horizontal distance from the centre line, it is the horizontal distance from the centre line to the inflexion point in the subsidence basin.

Alternatively, O’Reilly and New (1982), based on monitoring data from several tunnels in the UK, were able to prove that the horizontal surface displacements occur in the transverse direction of the excavation axis and, assuming that the resulting displacement vector is oriented towards the tunnel, the horizontal movement can be expressed as follows:

$$\:{S}_{hx}=\:\frac{{x\:S}_{v}\left(x\right)}{{z}_{0}}$$
(2)

where z0 depth of the tunnel or drift centre line, Shx is the horizontal movement at displacement distance x from the tunnel centre line.

The horizontal displacement corresponds to the inflection point of the subsidence basin. The horizontal deformation can be calculated by deriving the aforementioned expression (Eq. 2):

$$\:{\varepsilon\:}_{hx}\left(x\right)=\frac{{S}_{v}\left(x\right)}{{z}_{0}}\left(\frac{{x}^{2}}{{i}_{x}^{2}}-1\right)$$
(3)

Where the i parameter is the inflection point of the subsidence basin, \(\:{\varepsilon\:}_{hx}\) is the strain or horizontal deformation, and \(\:{S}_{v}\) is the surface subsidence at offset distance x from the tunnel centerline.

Fig. 9
figure 9

Distribution of horizontal strain, surface displacements, boundary angle and surface vertical settlements trough (based on O´Reilly and New 1982)

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In subsidence engineering, the terminology “deformation or strain (ε)” is the change in length over a piece of ground, expressed either as a dimension over the whole length or as a fraction of the unit of length. The direction is always specified with extensions and compressions, indicated by a + and – sign, respectively. Furthermore, the degree to which any surface site may be expected to tilt as a result of subsidence is calculated from the subsidence profile. Prediction of deformation from curvature is a useful tool that can be applied to any part of any profile.

The curvature can be calculated by dividing the subsidence difference by the distance between the observed stations (STN), which gives the slope (θ), determining the curvature and the strain. An example can be seen in Fig. 10.

Fig. 10
figure 10

Example of strain curvature (based on National Coal Board (1975)

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Results and discussion

The following sections describe the results obtained. They present the division between active and residual subsidence areas, characteristic boundary angles, and distances of influence. In addition, they provide the characteristic parameters of the subsidence basin function obtained from fitting the curve to a Gaussian distribution using the least-squares method.

Active and residual areas in the Catalan Potash Basin (E-CPB)

The methodology enabled the comparison of several subsidence profiles over a decade. Therefore, detecting the progress of active and residual subsidence areas was possible. Four subsidence areas were determined in the E-CPB and classified according to the cardinal directions. Three of them, the northern, the southwestern, and the southern regions, were defined as active areas. In contrast, the eastern area was considered to be in a residual subsidence process, given that there has been no mining activity since 2009. Non-cumulative profiles were used to confirm the tendency. An example of this trend is displayed in Fig. 11. The sequence of profiles A-A’, B-B’, C-C’ and D-D’, all from the period of 2020–2021, shows that the subsidence basin is stable in profiles located in the eastern area, therefore, categorising them into a residual subsidence process in contrast to the profiles A-A’ and B-B’ which belong to an active one.

Fig. 11
figure 11

(a–d) Sequence of profiles A-A’, B-B’, C-C’ and D-D’, from 2020 to 2021

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Characteristic boundary angle and distance of influence

The procedure discussed earlier achieved the criteria of boundary angle and distance of influence. Previous studies determined a general value of the subsidence angle of 35º (Sanmiquel et al. 2018). However, the proposed new approach allowed the study of the specific areas of the subsidence basin by means of 81 profiles from 12 different periods. In this regard, three values for the characteristic boundary angle and distance of influence have been calculated. In Fig. 12, a graphical scheme of the target area (shown in red), the selected sections (demonstrated in pink), the characteristic average boundary angles \(\:\left(\underline{\alpha\:}\right)\), and distance of influence \(\:\left(\underline{d}\right)\) are displayed (highlighted in green, blue and yellow). The highest value of the boundary angle is 71˚ in the northern zone, reaching its minimum value, 38˚, in the southern zone. The distance of influence is inversely proportional to the boundary angle trend, and, therefore, it experiences a value of 240 m in the north area while it soared to 988 m in the south. Furthermore, as presented in the 3.1 section, the southeast zone corresponds to a stabilised subsidence basin (highlighted in grey).

A possible explanation for the variation of the boundary angle and the distance of influence could stem from the area’s geological structure. The case study (Section “Case study”) showed that the area was folded due to alpine orogeny forces. Considering Fig. 13, in the northwestern part of the study area, a thrust fault between the anticlinal and syncline folds can be seen, which could be one of the factors that cause the north and west boundary angles to be greater.

Another factor that might cause the difference in boundary angles and distances of influence might be the design of the mining drifts. In contrast, in the northern part of the mine, two mine drifts were used to extract layers A and B, and in the southern part of the ore deposit, only one mine drift was required, as the salt layer between the two potash beds had a thicker less than 5 m. However, other mining and geological factors might also affect it, as it is stated that the rate of exploitation or the depth at which the mining galleries are located can also be key factors in the subsidence process (Hunt 1980; Sahu et al. 2017).

The main limitation of the study is the lack of understanding of the reason for the difference between boundary angles and influence distances. Although it can be seen that the geological structures or the mining designs used in the study area could be influencing parameters, future studies will be required to confirm which has the most influence on the subsidence that occurred in the CPB.

Fig. 12
figure 12

Average boundary angle and distance of influence of the active zones and the profiles used for the analysis

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Fig. 13
figure 13

Geological map of the northwest part of the study area (E-side in Fig. 1a), where a thrust fault can be identified between the anticline and syncline fold

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Approximation of the characteristic function of the subsidence basin

In the study by Sanmiquel et al. (2018), four sections were analysed beyond the zone of influence from 2008 to 2016; sixteen cross-settlement profiles were measured in the four sections. However, with this new approach, it has been possible to gather more subsidence data, as nine profiles have been defined in the period 1995 and 2020 (see Figs. 6 and 10), providing 99 transverse settlement profiles, of which 74 have been analysed, this represents a more accurate approach than in the previous study. Additionally, these results have allowed us to compare them with the data published by Sanmiquel et al. (2018), and consequently, the tuning of the proposed algorithm has been improved. An example of the transverse settlement profiles 1–1’ from 1995 to 2021 can be seen in Fig. 14.

Fig. 14
figure 14

Surface transverse settlement profiles in the section 1-1´

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As in the London clay materials, in the saline materials, the shape of the surface subsidence profiles are reasonably well represented by a Gaussian distribution, Eq. 1 (Mair et al. 1993). The Gaussian curve was successfully fitted to the field data using the least square method (Fig. 15).

Fig. 15
figure 15

Interpretation of measurements by an empirical Gaussian curve. (a) Section 1_1’_2016_2017 (b) Section 1_1’_2017_2018

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The width of the settlement profile is defined by the important parameter (i), which is the distance from the centerline of the trenches to the inflexion point of the trough (shown in Fig. 15). This parameter has been obtained from the Gaussian curve adjusted to the field data, obtaining a value of I for each transverse cross-section of the subsidence profiles. Seventy-four values of parameter “i” were obtained. From the values of parameter i, the horizontal deformation was calculated following Eq. 3.

The average values of the strains are proportional to θ/l, with horizontal deformation, have been determined within reasonable limits of accuracy and are shown in the prediction graph (Fig. 16). This figure provides a quantified relationship between deformation and θ/l, as follows (Eq. 4).

$$\:\frac{\theta\:}{l}=\frac{{\left(deformation\right)}^{2}}{x}$$
(4)

x (parameter dimensionless) varies between 0.018 and 0.012 from the profiles shown in 2018 (θ/l), which gives the differential slope between two values with subsidence data.

In this case, a new algorithm has been fitted more accurately to all data between 1995 and 2021, obtaining all values in the proposed range in 2018 (0.018 and 0.012) but with a higher concentration of x values between (0.015 − 0.013). Eventually, it can be said that a new algorithm was successfully obtained, according to the field data. Therefore, the new algorithm calculates the horizontal ground deformations at any point of the subsidence basin. In addition, based on the recently calculated limit, a lower deformation value is suggested.

Fig. 16
figure 16

Relationship between strain and θ/l meters

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Conclusions

The research introduces a novel approach to characterise and predict subsidence basins in the area of interest. This new methodology allows for efficiently handling large datasets to determine key parameters of any subsidence basin, with potential applications in other subsidence basin case studies. The analysis of 74 subsidence profiles has been done based on the methodology presented. Thus, the main findings of the case study are the following:

  1. 1.

    The area of interest has been divided into four zones according to cardinal directions: North, South, East, and Southwest.

  2. 2.

    The eastern zone is considered under a process of residual subsidence, while the other three belong to an active subsidence process. In that case, it has been possible to calculate their boundary angle and distance of influence.

  3. 3.

    The highest boundary angle value is 71˚ in the northern zone, decreasing towards the south. Thus, in the southwest part, it reaches a value of 52˚ and eventually reaches its minimum value, 38˚, in the southern zone.

  4. 4.

    The distance of influence is inversely proportional to the boundary angle trend; therefore, it has the smallest value in the northern part with 240 m, the southwestern part has a value of 624 m, and the maximum value is 988 m in the southern area.

In addition, the presented methodology allows using the least squares method to approximate the subsidence curve to the Gaussian function successfully. Moreover, the characteristic parameters, such as the key parameters “i”,” x”, and “θ/l”, were identified following the same methodology. The subsidence process algorithm has also been accurately calculated, allowing for a precise approximation of deformation at any given point in the area of interest. This enhancement significantly improves the forecasting and prediction of the subsidence basin, resulting in increased safety levels in the mining area and its surroundings.

Finally, the presented method can constitute an appropriate complement to upgrading the management of land subsidence in mining companies, not only for its adaptability and simplicity but also due to its effective, accurate, and functional approach.